Communications in the Observation Limited Regime by Manish Bhardwaj B.A.Sc. (Honors), Nanyang Technological University (1997) S.M., Massachusetts Institute of Technology (2001) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science and Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2009 c Massachusetts Institute of Technology 2009. All rights reserved. ! Author............................................. ................. Department of Electrical Engineering and Computer Science May 22, 2009 Certified by......................................... ................. Anantha Chandrakasan Joseph F. and Nancy P. Keithley Professor of Electrical Engineering Thesis Supervisor Accepted by . ................ Terry P. Orlando Chairman, Department Committee on Graduate Theses 2 Communications in the Observation Limited Regime by Manish Bhardwaj Submitted to the Department of Electrical Engineering and Computer Science on May 22, 2009, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science and Engineering Abstract We consider the design of communications systems when the principal cost is observ- ing the channel, as opposed to transmit energy per bit or spectral efficiency. This is motivated by energy constrained communications devices where sampling the signal, rather than transmitting or processing it, dominates energy consumption. We show that sequentially observing samples with the maximum a posteriori entropy can re- duce observation costs by close to an order of magnitude using a (24,12) Golay code. This is the highest performance reported over the binary input AWGN channel, with or without feedback, for this blocklength. Sampling signal energy, rather than amplitude, lowers circuit complexity and power dissipation significantly, but makes synchronization harder. We show that while the distance function of this non-linear coding problem is intractable in gen- eral, it is Euclidean at vanishing SNRs, and root Euclidean at large SNRs. We present sequences that maximize the error exponent at low SNRs under the peak power con- straint, and under all SNRs under an average power constraint. Some of our new sequences are an order of magnitude shorter than those used by the 802.15.4a stan- dard. In joint work with P. Mercier and D. Daly, we demonstrate the first energy sam- pling wireless modem capable of synchronizing to within a ns, while sampling energy at only 32 Msamples per second, and using no high speed clocks. We show that tradi- tional, minimum distance classifiers may be highly sensitive to parameter estimation errors, and propose robust, computationally efficient alternatives. We challenge the prevailing notion that energy samplers must accurately shift phase to synchronize with high precision. Thesis Supervisor: Anantha Chandrakasan Title: Joseph F. and Nancy P. Keithley Professor of Electrical Engineering Acknowledgments I am indebted to Anantha for his support, encouragement, the freedom to pursue problems, and the collaborative lab environment from which this work emerged. I am grateful to my committee members Dave Forney and Lizhong Zheng for their feedback and encouragement. An extra thank you to Dave, who taught me commu- nications design and coding. I am also thankful to Anant Sahai at Berkeley for his encouragement in pursuing the observation limited setting. Thanks to my collaborators Pat and Denis! I learnt from both of them, and the thesis is better as a result. Thanks to Vivienne, Maryam, Rana, Amrita and Naveen for being willing movie and dinner partners. Thanks to the many friends who reviewed or helped with this work, including Joyce, Julius, Maryam, Raj and Sashi. Thanks to Rex, Eugene, and Raj — my first friends at MIT! Thank you Kush and Lindi, and Kathleen! Thank you Bill, for your deep commitment to public service, and grinding it out in Bihar! I would not have made it without Creighton’s unwavering friendship, or the love and encouragement of my brother Ashu. How do I thank my parents, Renu and Manoj, for their sacrifices, hard work and love that went in raising Ashu and me? I hope we can lead lives worthy of their love and toil. This work was funded by DARPA HI-MEMS program (Contract # FA8650-07-C-7704), and in part an IBM Research Fellowship, and a grant from the NSF. 6 Contents 1 Introduction 15 1.1 ModelingEnergyConsumption . 18 1.1.1 TransmitversusReceiveEnergy . 18 1.1.2 Sampling versus Processing Energy . 19 1.1.3 Circuit Startup Time . 19 1.2 ThesisContributions ........................... 20 2 Coding Under Observation Constraints 23 2.1 Preliminaries . 23 2.2 Fundamental Limits, Relation to Feedback . 24 2.3 Traditional Coding . 25 2.4 TheSPRTwithUncodedData ..................... 25 2.5 Sampling Block Codes When r 0.................. 28 rx → 2.5.1 Chernoff’sProcedureA...................... 29 2.5.2 Application to Sampling Codes . 30 2.6 Sampling Block Codes When rrx ! 0.................. 31 2.6.1 Alternatives to Procedure A . 31 2.6.2 Application to Sampling Codes . 33 2.7 Examples of Sampling to Reduce Receive Cost . 34 2.8 PracticalConsiderations . 39 2.8.1 VLSI Costs . 39 2.8.2 ImpactofNon-ZeroTransmitRates. 43 2.9 ComparisonswithCurrentReceivers . 44 7 2.10Summary ................................. 46 3 Synchronization by Sampling Energy 47 3.1 Preliminaries . 49 3.1.1 Energy Sampling Channel, Integrated Codebooks . 49 3.1.2 Synchronization Codes . 51 3.1.3 Elementary Distance Properties . 54 3.1.4 Performance Analysis . 56 3.2 PriorWork ................................ 58 3.2.1 Periodic Sequences . 59 3.2.2 OversampledSequences . .. .. 61 3.3 InterpolatedSequences .......................... 64 3.4 Maximum Squared Distance Sequences . 71 3.4.1 Simple, Strong Sequences . 71 3.4.2 Non-Simple MSDs . 74 3.4.3 Distance Comparison . 76 3.5 WalkingSequences ............................ 77 3.5.1 Distance Comparison . 85 3.5.2 Almost-block w-sequences .................... 86 3.6 Simulation Results . 86 3.6.1 Low SNR . 88 3.6.2 High SNR . 90 3.6.3 SummaryofImprovements. 92 3.7 Average Versus Peak Power Constraint . 94 3.8 Summary ................................. 95 3.9 SelectedProofs .............................. 97 3.9.1 ChernoffInformation of χ2 r.v.s as SNR 0, . 97 → ∞ 3.9.2 Bound on Phases in a Simple, Strong MSD Sequence . 104 4 An Energy Sampling Wireless Modem 109 4.1 Overview.................................. 110 8 4.1.1 Pulsed Ultra-Wideband Signaling . 110 4.1.2 PacketFormat .......................... 110 4.1.3 Modem Units . 112 4.2 VLSI EfficientClassifiers......................... 113 4.2.1 Suitable Approximations . 113 4.2.2 Reducing Computation . 117 4.2.3 Distributing Computation . 119 4.3 Chip Details . 120 4.4 Measurements............................... 121 4.5 Summary ................................. 122 5 Conclusions 125 5.1 OpenProblems .............................. 125 5.1.1 Coding............................... 125 5.1.2 InformationTheory. .. .. 126 5.1.3 SequenceDesign ......................... 127 5.1.4 VLSI . 129 5.2 Outlook .................................. 130 A Tables of Sequences with Good Squared Distance 133 B Tables of Walking Sequences 139 B.1 Block w-sequences ............................ 139 B.2 Almost-block w-sequences ........................ 140 9 10 List of Figures 1-1 Setup of the coding under observation cost problem. 16 1-2 Structure of a wireless transmitter and a non-coherent receiver. 18 2-1 The SPRT as decoder for uncoded transmission over the binary AWGN channel. .................................. 27 2-2 Receive costs of the rate 1/2 SPRT and K = 7 convolutional code. 27 2-3 Performance of the rrx =1/2, (7,3) code using ME and uniform sam- pling(U). ................................. 35 2-4 Performance of the rrx =1/10, (7,3) code using ME and uniform sam- pling..................................... 36 2-5 Performance of the Golay code at rrx =1/2. .............. 37 2-6 Sampling the Golay code with receive rate 1/6. 38 3-1 The discrete-time, energy sampling channel . 50 3-2 Performance in the low SNR regime with (a) P =2, and each sequence being repeated 16x, and, (b) P =16, and every sequence repeated 64x. 90 3-3 Performance in the high SNR regime (P =2)............... 91 3-4 Performance for P =16 in the high SNR regime. 93 4-1 Packet format used in energy sampling modem. Pulse phase scrambling andmodulationarenotshown. 111 4-2 Block diagram of the energy sampling modem. 112 11 4-3 The ML, UVG, and MD classifiers for a (31,16) I-H sequence repeated 32x and 1x. The UVG curve lies between ML and MD (32x), and overlapswithML(1x). .......................... 115 4-4 Impact of SNR mismatch on the MD and UVG classifiers. Dotted curves denote no mismatch. Curves on the left and right are for 32 and 1 sequence repetition, respectively. 116 4-5 A single-chip energy sampling modem. 121 4-6 Synchronization error rate measured on the chip (courtesy Patrick Mercier), compared with an unquantized ML implementation (sim- ulated). SNR is the ratio of average power to noise, and not SNR per symbol. .................................. 122 5-1 WhatdoestheParetofrontierlooklike? . 129 12 List of Tables 2.1 Computation required per sample for a brute-force implementation of the ME algorithm. We use Π to denote the vector of the M =2k codeword posteriors, and i∗ for the index of the observed bit. 40 3.1 Best known squared-distances for given nr,P. Sequences in bold are strong. Subscripts indicate sequence dimension. All sequences without a subscript are simple. (†No new sequences are listed for nr=63 and P<8. The distance achieved by I-H sequences is quoted instead.) . 76 3.2 Block w-sequences for arbitrary P P (t), conjectured to achieve a ≥ lim distance of 2t. Only non-identity blocks are specified. 85 3.3 Almost-block w-sequences conjectured to achieve a distance of 2t (ver- ified for P 16). The first block has length P + 3 and the rest P + 2. ≤ Only non-identity blocks are specified. Sequences for P values not included here are in appendix B.2. 87 3.4 Bounds on sample length for an integer distance of 2t, versus search complexity (number of candidate sequences) of various families.